Method for filtering seismic data, particularly by kriging

ABSTRACT

A method of filtering at least two series of seismic data representative of the same zone, the method including the step of determining an estimate of the component that is common to the data series, and the step of deducing from this estimate a resolution of the data series. The method is applicable to 4D seismic surveying.

GENERAL TECHNICAL FIELD—BACKGROUND ON KRIGING ANALYSIS

The present invention relates to filtering seismic data, in particularby kriging analysis.

Kriging analysis enables a random function to be resolved from itscovariance function.

In particular, it is conventionally used in geostatistics for filteringseismic data, particularly but in non-limiting manner, forcharacterizing reservoirs.

Kriging analysis relies in particular on the assumption that aphenomenon measured locally by means of optionally-regular sampling canbe analyzed as a linear sum of a plurality of independent phenomena, thevariogram of the overall phenomenon corresponding to the linear sum ofthe variograms of each of the independent phenomena making it up.

Conventionally, the variogram corresponding to the measured experimentaldata is resolved as sum of modeled variograms, and from the experimentaldata and the models selected for the individual variograms used toresolve the data, there are deduced the individual functions that makeup the random function corresponding to the overall phenomenon.

It is thus possible to extract from a seismic data map of the type shownin FIG. 1 (e.g. raw experimental data) firstly the white noise that ispresent in the data (FIG. 2 a), secondly noise corresponding to linearlines (FIG. 2 b), and finally filtered data cleared of both of thesekinds of noise (FIG. 2 c).

The kriging calculations for determining the values of the individualfunctions into which an overall random function is resolved arethemselves conventionally known to the person skilled in the art.

In this respect, reference can be made, for example, to articles andpublications mentioned in the bibliography given at the end of thepresent description.

Very generally, the value of an individual function involved in makingup the overall random function is determined as being a linearcombination of experimental values for points in an immediateneighborhood of the point under consideration, these experimental valuesbeing given weighting coefficients.

In other words, if it is considered that a function Z(x) is made up asthe sum of individual functions Y^(u)(x), this can be written:

${Z(x)} = {\sum\limits_{u = 1}^{U}{Y^{u}(x)}}$and the component Y^(u)(x) is estimated by:

${Y^{u_{*}}(x)} = {\sum\limits_{\alpha = 1}^{N}{\lambda_{u}^{\alpha}Z_{\alpha}}}$where α is a dummy index designating the points under considerationaround the point x for which it is desired to determine the estimatedvalue Y^(u)*(x), Z_(x) being the value at the point x, N being thenumber of such points.

It can be shown that the weighting coefficients λα satisfy the equation:

${\begin{pmatrix}C_{11} & \ldots & C_{1N} \\\vdots & \; & \vdots \\C_{N1} & \ldots & C_{NN}\end{pmatrix}\begin{pmatrix}\lambda_{u}^{1} \\\vdots \\\lambda_{u}^{N}\end{pmatrix}} = \begin{pmatrix}C_{01}^{u} \\\vdots \\C_{0N}^{u}\end{pmatrix}$where the index 0 designates the point for which an estimate is to bedetermined, the values C₀₁ ^(u) to C_(0N) ^(u) being the covariancevalues calculated from the model u corresponding to the component Y^(u)(values of the covariance function for the distances between each datapoint and the point to be estimated), the values C_(ij) being covariancevalues calculated as a function of the selected model for the variogramof the function to be estimated (values of the covariance function forthe distances between the data points).

These weighting coefficients λ_(u) ^(x) are thus determined merely byinverting the covariance matrices.

PROBLEMS POSED BY THE STATE OF THE ART—SUMMARY OF THE INVENTION

One of the difficulties of presently-known kriging analysis techniquesis that they require the use of models of the covariance functions.

The advantage of using such models is that they make it possible to havematrices which are defined, positive, and invertible.

Nevertheless, it will be understood that although such filteringtechniques give good results, they are strongly dependent on theindividual expertise of the person selecting the models for the variousvariograms.

That can be a source of error, and prevents those techniques being usedby people who are not specialists.

Furthermore, selecting models also leads to significant losses of timein production.

An object of the invention is to mitigate that drawback and to propose afiltering technique using kriging analysis that can be implemented inautomatic or almost automatic manner.

The invention provides a method of filtering at least two series ofseismic data representative of the same zone, by determining (e.g. bydetermining the cross variogram of the data series and solving theco-kriging equation) an estimate of the component that is common to thedata series, and deducing a resolution of these data series is deducedfrom the estimate.

The invention also provides a method of processing seismic data in whicha filter method of the above-specified type is implemented in order tocompare two series of seismic data corresponding, for the same zone, togrids of at least one common attribute obtained for two distinct valuesof at least one given parameter.

The invention also provides a method of filtering at least one series ofdata representative of the values of at least one physical parameterover at least one zone, characterized by identifying a model of acomponent of three-dimensional variability of its variogram, subtractingsaid model from the experimental variogram, and solving the krigingequation corresponding to the different variograms in order to deduce anestimate of the corresponding variability component on the data series.

BRIEF DESCRIPTION OF THE FIGURES

Other characteristics and advantages of the invention appear furtherfrom the following description which is purely illustrative andnon-limiting and should be read with reference to the accompanyingdrawings, in which:

FIGS. 1 and 2 a, 2 b, and 2 c, described above, illustrate an example ofseismic data mapping and of the corresponding resolution by kriginganalysis;

FIGS. 3 a and 3 b show two maps of the same zone, obtained fromacquisitions undertaken at two different times;

FIG. 4 is a map of the component that is common to the maps of FIGS. 3 aand 3 b;

FIGS. 5 a and 5 b and FIGS. 6 a and 6 b are maps of components otherthan the component that is common to the maps of FIGS. 3 a and 3 b; and

FIGS. 7 a to 7 c are graphs showing the distribution of errorsrespectively when using standard filtering, filtering by conventionalfactorial kriging, and filtering as proposed by the invention (factorialco-kriging).

DESCRIPTION OF ONE OR MORE IMPLEMENTATIONS OF THE INVENTION

Automatic Filtering

It is assumed that two maps are available that have been obtained forthe zone with seismic data acquired, for example, at different instantsor for seismic attributes that are different.

By way of example, these two maps are of the type shown in FIGS. 3 a and3 b.

The two functions corresponding to these two data series are written Z1and Z2 below.

It is proposed to resolve each of these two functions into the sum oftheir common component plus orthogonal residues.

For this purpose, there is determined, from two data series for which across variogram is available having the following values:

${\gamma_{12}(h)} = {\frac{1}{N}{\sum{\left( {{Z\; 1(x)} - {Z\; 1\left( {x + h} \right)}} \right)\left( {{Z\; 2(x)} - {Z\; 2\left( {x + h} \right)}} \right)}}}$where x and x+h designate the pairs of points taken into considerationin the direction and for the distance h for which the value of thevariogram is determined, and where N is the number of pairs of pointsfor said direction and said distance.

Knowing this cross variogram, an estimate is then determined of thefunction corresponding thereto, which satisfies:

${Z_{12}^{*}(x)} = {{\sum\limits_{\alpha = 1}^{N}{\lambda_{\alpha}^{1}Z_{\alpha}^{1}}} + {\sum\limits_{\beta = 1}^{N}{\lambda_{\beta}^{2}Z_{\beta}^{2}}}}$where α and β are two dummy indices designating the points taken intoconsideration around the point x for which it is desired to determine anestimate of said function, Z_(α) ¹ and Z_(β) ² being the value at saidpoint x, N being the number of said points, and where λ_(α) ¹ and λ_(β)² are weighting coefficients.

These weighting coefficients λ_(α) ¹ and λ_(β) ² are determined byinverting the co-kriging equation:

${\begin{bmatrix}{C_{11}11} & \ldots & {C_{11}N\; 1} & {C_{11}11} & \ldots & {C_{11}11} \\\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\{C_{11}1N} & \ldots & {C_{11}{NN}} & {C_{12}11} & \ldots & {C_{12}{NN}} \\{C_{21}11} & \ldots & {C_{21}N\; 1} & {C_{22}11} & \ldots & {C_{22}N\; 1} \\\ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\{C_{21}1N} & \ldots & {C_{21}{NN}} & {C_{22}11} & \ldots & {C_{22}{NN}}\end{bmatrix}\begin{bmatrix}\lambda_{11} \\\ldots \\\lambda_{1N} \\\lambda_{21} \\\ldots \\\lambda_{2N}\end{bmatrix}} = \begin{bmatrix}{C_{11}1X} \\\ldots \\{C_{11}{NX}} \\{C_{12}1X} \\\ldots \\{C_{12}{NX}}\end{bmatrix}$where the coefficients C12αβ and C21αβ are the cross-variance values ofthe functions Z1 and Z2 at the points corresponding to indices α and β,and where the coefficients C11αβ and C22αβ are the covariance valuesrespectively of the function Z1 and of the function Z2 at said points.The index X corresponds to the point referred to above as x.

It should be observed that the matrix which appears in this equation hasthe advantage of being invertible under certain calculation conditions.

In this way, using experimental covariances, the two variablescorresponding to the two initial data series are resolved automaticallyinto a common component and into two residual orthogonal components. Theregularity of the data means that the experimental covariance is knownfor all of the distances used, so no interpolation is needed, so thematrix is defined positive.

The function then obtained is an estimate of the component that iscommon to both data series.

FIG. 4 shows an estimate of the common component obtained from the datacorresponding to the maps of FIGS. 3 a and 3 b.

It will be understood that it is particularly advantageous in numerousapplications, and in particular in 4D seismic surveying, to have thiscommon component available.

It serves in particular to eliminate non-repeatable acquisitionartifacts:

-   -   on the basis of seismic attribute grids, and in particular, for        example, on the basis of root mean square (rms) amplitudes in an        interval;    -   on the basis of seismic time grids, and for example on the basis        of seismic event times; and    -   on the basis of seismic velocity volumes, and for example on the        basis of automatically-determined points of origin for velocity        vectors.

It may also be used in simple seismic surveying to eliminatenon-repeatable acquisition artifacts:

-   -   in particular on the basis of seismic attribute grids calculated        on consecutive incidence classes, or indeed    -   on the basis of seismic attribute grids calculated on volumes        obtained by partial summing or converted waves.

Furthermore, once an estimate has been determined for the commoncomponent, it is possible to determine the residual componentscorresponding to the difference between the initial data and saidestimated common component.

These residual components can themselves be resolved by kriginganalysis.

This is shown in FIGS. 5 a, 5 b and 6 a, 6 b which are maps of whitenoise and of linear line noise estimated in this way for each of the twoseries of measurements shown in FIGS. 3 a and 3 b.

Examples of error measurements as obtained by standard filtering, bykriging analysis filtering, and by co-kriging analysis filtering (ormultivariable kriging), are illustrated by the graphs of FIGS. 7 a to 7c.

On reading these figures, it will be understood that filtering byco-kriging analysis makes it possible to obtain dispersions that aremuch smaller than those obtained with conventional filtering orfiltering by kriging analysis, and gives better results.

The above description relates to an implementation using two series ofdata.

As will be readily understood, the proposed method can also beimplemented in the same manner with a larger number of data series(campaigns).

Semi-Automatic Filtering

This second implementation also enables simplified resolving when onlyone seismic data series is available (function S1).

It assumes that a model of a component θm of the experimental variogramθ is previously available.

This model which is known beforehand is, for example, the model of anindependent component of the subsoil geology: white noise, stripes, etc.

Knowing this model of the component θm, there is deduced therefrom theresidual variogram corresponding to the difference between theexperimental variogram and this component θm.

Kriging analysis is then performed in order to determine firstly themodel component Sm and secondly, on the basis of the residual variogram,the orthogonal residue R1 such that:S1=Sm+R1

This automatic resolution enables acquisition anomalies to be filteredout when they present three-dimensional coherence that is easilyidentified and modeled, such as the stripes parallel to the cables thatare observed in the amplitudes and times when performing offshoreseismic surveys.

REFERENCES

-   G. Matheron (1982), “Pour une analyse krigeante des données    régionalisées” [For kriging analysis of regionalized data], Internal    note, Center for Mathematical Morphology and Geostatiistics, Ecole    Nationale Supérieure des Mines, Paris.-   L. Sandjivy (1984), “Analyse krigeante des données    géochiliques—Etude d'un cas monovariable dans le domaine    stationnaire” [Kriging analysis of geochilic data-   Study of a single variable case in the steady domain], Sciences de    la Terre, série informatique géologique 18, pp. 143-172.-   A. Galli, L. Sandjivy (1984), “Analyse krigeante et analyse    spectrale” [Kriging analysis and spectral analysis], Sciences de la    Terre, série informatique géologique 21, pp. 115-124.-   P. Dousset, L. Sandjivy (1987), “Analyse krigeante des données    géochimiques multivariables prélevées sur un site stannifère en    Malaisie” [Kriging analysis of multivariable geochemical data taken    from a tin-bearing site in Malaysia], Sciences de la Terre, série    informatique géologique 26, pp. 1-22.-   L. Sandjivy (1987), “Analyse krigeante des données de prospection    géochimique” [Kriging analysis of geochemical prospection data],    Ph.D. thesis, engineer in mining techniques and sciences, ENSMP, 166    p.-   O. Jacquet (1988), “L'analyse krigeante appliquée aux données    pétrolières” [Kriging analysis applied to oil prospecting data],    Bulletin de l'Assoc. Suisse des Géologues et Ingénieurs du Pétrole,    Vol. 24, pp. 15-34.-   C. Daly (1989), “Application of multivariate kriging to the    processing of noisy images”, Geostatistics, Vol. 2, Kluwer Academic    Publisher, M. Armstrong (ed.), pp. 749-760.-   C. Daly (1991), “Applications de la géostatistique à quelques    problèmes de filtrage” [Applications of geostatistics to some    filtering problems], Ph.D. thesis, engineer in mining techniques and    sciences, ENSMP, 235 p.-   H. Wackernagel, H. Sanguinetti (1993), “Gold prospecting with    factorial kriging in Limousin, France”, Computers in Geology: 25    years of progress, Davis & Herzfeld (ed.), Oxford, O.U.P., Studies    in Mathematical Geology 5, pp. 33-43.-   S. Seguret (1993), “Analyse krigeante spatio-temporelle appliquée à    des données aéromagnétiques” [Spatio-temporal kriging analysis    applied to aeromagnetic data], Cahiers de Géostatistique, Fasc. 3,    ENSMP, pp. 115-138.-   H. Wackernagel (1998), “Multivariate geostatistics: an introduction    with applications”, 2nd ed. Berlin, Springer, 291 p.-   M. Arnaud et al., (2001), “L'analyse krigeante pour le classement    d'observations spatiales et multivariées” [Kriging analysis for    classifying three-dimensional and multivariate observations], Revue    de statistique appliquée, XLIX (2), pp. 45-67.

1. A method of filtering at least two series of seismic datarepresentative of the same subsurface zone, the method beingcharacterized by determining a cross variogram of these data series andsolving a co-kriging equation which results from this determination forautomatically deducing an estimate of the component that is common tothe data series, wherein each of the data series is resolved into thesum of said common component and its respective orthogonal residues,said resolution of the data series used for determining the topographyof the subsurface zone.
 2. A method according to claim 1, characterizedby determining the orthogonal residues for the various data series bysubtracting the estimated common component from each of the data series.3. A method according to claim 2, characterized by implementing kriginganalysis to resolve said orthogonal residues.
 4. A method of processingseismic data, comprising: comparing two series of seismic datacorresponding, for the same zone, to grids of at least one commonattribute obtained at two distinct values of at least one givenparameter, said comparing including filtering at least two series ofdata representative of the same subsurface zone by determining a crossvariogram of these data series and solving a co-kriging equation whichresults from this determination for automatically deducing an estimateof the component that is common to the data series, wherein each of thedata series is resolved into the sum of said common component and itsrespective orthogonal residues, said common component of the data seriesused for determining the topography of the subsurface zone.
 5. A methodaccording to claim 4, characterized by determining the orthogonalresidues for the various data series by subtracting the estimated commoncomponent from each of the data series.
 6. A method according to claim5, characterized by implementing kriging analysis to resolve saidorthogonal residues.
 7. A method of filtering at least one series ofseismic data representative of at least one subsurface zone, the methodbeing characterized by identifying a model of a component ofthree-dimensional variability of its variogram, subtracting said modelfrom the experimental variogram, and solving the kriging equationcorresponding to the different variograms in order to deduce an estimateof the corresponding variability component of the data series, whereinsaid estimate is used for determining the topography of the subsurfacezone.
 8. A method processing seismic data, comprising: comparing twoseries of seismic data corresponding, for the same subsurface zone, togrids of at least one common attribute obtained at two differentinstants, said comparing including filtering at least two series ofseismic data representative of the same subsurface zone by determining across variogram of these data series and solving a co-kriging equationwhich results from this determination for automatically deducing anestimate of the component that is common to the data series, whereineach of the data series is resolved into the sum of said commoncomponent and its respective orthogonal residues, said common componentof the data series used for determining the topography of the subsurfacezone.
 9. A method according to claim 8, characterized by determining theorthogonal residues for the various data series by subtracting theestimated common component from each of the data series.
 10. A methodaccording to claim 9, characterized by implementing kriging analysis toresolve said orthogonal residues.